CIRM - Videos & books Library - Blocks & gaps in the asymmetric simple exclusion process

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Abstract : In earlier work (arXiv:1707.04927) the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time t a particle is at site x and is the beginning of a block of L consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\rightarrow\infty$) that a particle is the beginning of an L-block, given that it is at site x at time t. Using duality between occupied and unoccupied sites we obtain the analogous result for a gap of G unoccupied sites between the particle at x and the next one.

Keywords : asymmetric simple exclusion process; Kardar-Parisi-Zhang scaling; cluster probabilities; step initial condition

MSC Codes :
82C20 - Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
82C22 - Interacting particle systems
82C23 - Exactly solvable dynamic models

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https://www.cirm-math.fr/RepOrga/2104/Slides/Tracy.pdf

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https://videos.cirm-math.fr/2019-04-11_Tracy.mp4

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Event Title : Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial Data
Dates : 08/04/2019 - 12/04/2019
Event Year : 2019
Event URL : https://www.chairejeanmorlet.com/2104.html

Citation Data

DOI : 10.24350/CIRM.V.19517303
Cite this video as: (2019). Blocks & gaps in the asymmetric simple exclusion process. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19517303
URI : http://dx.doi.org/10.24350/CIRM.V.19517303

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Bibliography

Craig A. Tracy and Harold Widom ; Blocks and gaps in the asymmetric simple exclusion process: Asymptotics - Journal of Mathematical Physics 59, 091401 (2018) - https://doi.org/10.1063/1.5021353

Amir, G., Corwin, I., and Quastel, J., “Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions,” Commun. Pure Appl. Math. 64, 466–537 (2011). - https://doi.org/10.1002/cpa.20347

Corwin, I., “The Kardar-Parisi-Zhang equation and universality class,” Random Matrices: Theory Appl. 1, 1130001 (2012), 76 pages. - https://doi.org/10.1007/s10955-015-1250-9

Sasamoto, T. and Spohn, H., “The crossover regime for the weakly asymmetric simple exclusion process,” J. Stat. Phys. 140, 209–231 (2010) - https://doi.org/10.1007/s10955-010-9990-z

Spitzer, F., “Interaction of Markov processes,” Adv. Math. 50, 246–290 (1970) - https://doi.org/10.1016/0001-8708(70)90034-4

Tracy, C. A. and Widom, H., “Integral formulas for the asymmetric simple exclusion process,” Commun. Math. Phys. 279, 815–844 (2008) - https://doi.org/10.1007/s00220-008-0443-3

Tracy, C. A. and Widom, H., “Erratum to: Integral formulas for the asymmetric simple exclusion process,” Commun. Math. Phys. 304, 875–878 (2011 - https://doi.org/10.1007/s00220-011-1249-2

Tracy, C. A. and Widom, H., “A Fredholm determinant representation in ASEP,” J. Stat. Phys. 132, 291–300 (2008) - https://doi.org/10.1007/s10955-008-9562-7

Tracy, C. A. and Widom, H., “Asymptotics in ASEP with step initial condition,” Commun. Math. Phys. 290, 129–154 (2009) - https://doi.org/10.1007/s00220-009-0761-0

Tracy, C. A. and Widom, H., “Blocks in the asymmetric simple exclusion process,” J. Math. Phys. 58, 123302 (2017) - https://doi.org/10.1063/1.4996345

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